Since outer fan-planar graphs have
a density of \(3n-5\)  [1]}, a tile is not
outer fan-planar. Note that every outer fan-crossing graph is outer
fan-planar, since an edge cannot be crossed from both sides if all
vertices are in the outer face. Thus there is no configuration II
[2]}. Hence, in every fan-crossing embedding, each
tile \(T\)  has at least one vertex in its interior, called the
center of \(T\) , and at most 24 edges in the boundary of its
outer face. There are at least three vertex-disjoint paths between
the center and vertices in the boundary, since a tile is
3-connected.
